We compute the weak bending angle of light within generalised Born–Infeld electrodynamics as it passes through the equatorial plane of a magnetic dipole. We start by considering the refractive index associated with the dipole within generalised Born–Infeld electrodynamics. Then, we calculate the Gaussian optical curvature based on these refractive indices. Using the Gauss–Bonnet theorem, we derive a formula to quantify the deflection angle in the presence of a strong magnetic field from a dipole. Our results align with results obtained through traditional geometric optics techniques, underscoring the importance of the Gauss–Bonnet theorem as a versatile tool for solving intricate problems in modern theoretical research. We apply our theoretical deflection angle formula to estimate the light bending in magnetars listed in the McGill catalogue, providing insights into the behaviour of light in environments with strong magnetic fields.

In this article, we provide a review of the current state of the research on the nonlinear electrodynamics of vacuum in the presence of strong electromagnetic fields. We discuss the parametrized post-Maxwellian formalism that can be facilitate calculations in the weak electromagnetic field approximation. The post-Maxwellian parameters are a set of parameters used in non-linear electrodynamics to describe the behavior of electromagnetic fields in the presence of strong fields. We also discuss the equations for the electromagnetic field in the Born-Infeld electrodynamics and in non-linear Heisenberg-Euler electrodynamics, as well as the exact solution in the form of a plane elliptically polarized wave. We represent the Lagrangian of nonlinear electrodynamics in vacuum in detail. We demonstrate that the Lagrangian can be expressed as a series of integer powers of invariants J1 and J2, which allows for the identification of nonlinear electrodynamics that are consistent with experiments carried out in a weak electromagnetic field.

We consider a model problem of polarization-dependent light bending and time delays in the vicinity of neutron stars endowed with magnetar-strength magnetic fields (B∼${10}^{15}\mathrm{G}$), combining an effective-metric formulation of Heisenberg–Euler nonlinear electrodynamics with general-relativistic ray tracing. The spacetime geometry is analyzed using both the Kerr metric and a quadrupole-deformed q-metric, characterized by a quadrupole parameter varying in the range $q\in [{10}^{-3},0.5]$. In addition, the impact of complex magnetic-field topologies is examined by introducing a magnetic quadrupole component alongside the dipole configuration. The simulations performed in this study demonstrate that the inclusion of the quadrupole deformation parameter significantly modifies photon trajectory deflections compared to the standard Kerr solution. We further quantify the geometric dilution of the photon beam, finding a cross-section expansion ratio of approximately $4.7\times {10}^{13}$ for rays reaching Earth. This strong dilution imposes stringent constraints on the detectability of polarization-dependent signatures and time-delay echoes. Finally, characteristic illustrations are presented for trajectory distortions, bending-angle distributions, and intensity valleys produced by the combined gravitational and magnetic lensing effects.